Mathematics for the interested outsider

The Algebra of Differential Forms

We’ve defined the exterior bundle over a manifold . Given any open we’ve also defined a -form over to be a section of this bundle: a function such that . We write for the collection of all such -forms over . It’s straightforward to see that this defines a sheaf on .

This isn’t just a sheaf of sets; it’s a sheaf of modules over the structure sheaf of smooth functions on . We define the necessary operations pointwise:

where the right hand sides are defined by the vector space structures on the respective .

We can go even further and define the sheaf of differential forms

This sheaf is not just a sheaf of modules over , it’s a sheaf of algebras. For an and a , we define their exterior product pointwise:

In fact, this is a graded algebra, and the multiplication has degree zero:

Even better, this is a unital algebra. We see this by considering the zero grade, since the unit must live in the zero grade. Indeed, , so sections of are simply functions on . That is, . Given a function we will just write instead of .

[…] looks sort of familiar as a derivative, but we have another sort of derivative on the algebra of differential forms: the “exterior derivative”. But this one doesn’t really look like a derivative at […]

[…] smooth maps, and homotopies form a 2-category, but it’s not the only 2-category around. The algebra of differential forms — together with the exterior derivative — gives us a chain complex. Since pullbacks of […]

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This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.